Problem: Simplify the following expression and state the condition under which the simplification is valid. You can assume that $r \neq 0$. $t = \dfrac{9(5r - 7)}{-6} \div \dfrac{5r(5r - 7)}{6r} $
Explanation: Dividing by an expression is the same as multiplying by its inverse. $t = \dfrac{9(5r - 7)}{-6} \times \dfrac{6r}{5r(5r - 7)} $ When multiplying fractions, we multiply the numerators and the denominators. $t = \dfrac{ 9(5r - 7) \times 6r } { -6 \times 5r(5r - 7) } $ $ t = \dfrac{54r(5r - 7)}{-30r(5r - 7)} $ We can cancel the $5r - 7$ so long as $5r - 7 \neq 0$ Therefore $r \neq \dfrac{7}{5}$ $t = \dfrac{54r \cancel{(5r - 7})}{-30r \cancel{(5r - 7)}} = -\dfrac{54r}{30r} = -\dfrac{9}{5} $